The Hochschild cohomology ring of a group algebra is an object that has received recent attention, but is difficult to compute, in even the simplest of cases. In this paper, we use the product formula due to Witherspoon and Siegel to extend some of t
heir computations. In particular, we compute the Hochschild cohomology algebra of group algebras kG where |G| is less than 16, and we provide an alternative computation of the ring $HH^*(k(E ltimes P))$ considered by Kessar and Linckelmann.
Motivated by recent problems regarding the symmetry of Hecke algebras, we investigate the symmetry of the endomorphism algebra $E_P(M)$ for $P$ a $p$-group and $M$ a $kP$-module with $k$ a field of characteristic $p$. We provide a complete analysis f
or cyclic $p$-groups and the dihedral 2-groups. For the dihedral 2-groups, this requires the classification of the indecomposable modules in terms of string modules and band modules. We generalize our techniques to consider $E_{Lambda}(M)$ for $Lambda$ a Nakayama algebra, a local algebra, or even an arbitrary algebra.
